Suppose Bobby is cycling
east at 5m/s, and Lisa is cycling north at 10m/s happen to collide. They
and their bikes get all tangled up and start moving together. What's the
velocity of the Bobby-Lisa-bicycle glob right after the collision?
Let's take the combined mass of Bobby and his bike to be 
and
Lisa and her bike to be 
. (Don't worry, Bobby and Lisa are
fine. This is not a real examaple.)
To figure this out, we'll use conservation of momentum, as we claimed that to a good approximation, we can ignore other forces during the collision.
. where 
denotes
the final velocity, right after the collision.
Equating these two expressions (since momentum is conserved) we have
Now 
and 
so we get
The combined mass goes shooting off in the north-east direction.
This is an example of a totally inelastic collision. When the two masses hit, they stick together. The final velocity is just the center of mass velocity of the system, since the center of mass velocity is constant for any process obeying conservation of momentum.
Momentum is conserved but in general, energy is not. You could calculate the change in kinetic energy during this collision and would find that it is negative. What happens to this energy? It mostly goes into heat, some of it goes into sound waves. The scrunching sound that you hear is powered by the initial kinetic energy of our two unfortunate bicyclists.